Rebuilding a leaking or, dripping single handle faucet is usually a fairly easy task with the worst part being finding out who made it, and what parts you need along with where to find them. If you are able to identify the faucet manufacturer and model so you can view a parts diagram on-line you will be in much better shape than not having the information.
The first step in rebuilding the faucet is to shut off the water to the faucet then remove the handle. If you have the parts diagram you should consult that to see how the handle is attached and remove it accordingly. If you do not have the parts diagram you will need to figure out how the handle is attached. There may be a button covering the screw or a plug covering a access hole for the screw, some may even just pull off with a sharp tug, you will have to carefully examine the handle to determine how it comes apart. Some are pretty difficult with a small hole that you have to stick a screwdriver or allen wrench into in order to loosen a screw that secures the handle in place, often the handle has to be turned partially on before the screw can even become accessible.
Once the handle has been removed there are often some additional parts that have to be removed to access the water shutoff mechanism whether it is a cartridge or, a ball to remove it. Some will have parts that just slide off, others will have parts that unscrew without a diagram you will have to rely on your mechanical ability to guide you on how the parts come off. (A word of caution to people who are attempting to repair a Delta tub/shower faucet many Delta models have the valve in two sections that are connected by three small tubes. If you attempt to remove the bonnet nut and turn it too hard you can tear the tubes apart and ruin the valve.) Sometimes there is a nut that holds the cartridge or, in the case of Delta style kitchen sink faucets and some Delta style tub/shower faucets hold the ball in place, on Moen faucets there is a clip which secures the cartridge in place, other brands such as American Standard have screws holding the cartridge in place. In most cases the parts simply lift out while in others like the Moen you need to use a special cartridge removal tool to turn the cartridge to break it free then pull it out.
On kitchen faucets if there is leakage coming out from under the faucet spout the O-Rings under the spout are bad and need replacement. When I repair a kitchen faucet I replace the O-Rings and lubricate them with plumbers grease to prevent leaks and keep the spout moving easily. Once everything holding the spout down has been taken off the spout pulls off by pulling upward and turning the spout from side to side. Looking up the O-Ring size on the parts diagram is best but you can also take the existing O-Rings to match them.
Once you remove the cartridge or, in the case of many of the Delta style faucets the ball, rubber seats and springs along with the cam and packing you are ready to get the parts you need to rebuild the faucet. Many hardware stores, plumbing supply houses, and home centers will have a selection of cartridges you can match up, and everyone should have the parts for the Delta ball type faucet. If you cannot find a cartridge that matches your faucet Alfano Plumbing Parts has an excellent on-line catalog that you can look through to compare against your cartridge.
Once you have obtained the parts, assembly is pretty much just a reverse of how you pulled the faucet apart. Cartridges like the Moen that slide into a bore and seal with O-Rings should be lubricated with plumbers grease. The Delta ball and some cartridges have indexing pins that need to be lined up properly when reassembling. One of the things I have found helpful is when pulling things apart laying them out in order of assembly is often helpful on more complex faucets. On some of the older Delta ball style faucets there is a screw in the center of the cap that has to be adjusted for the correct pressure on the cam and packing. If the cam and packing are too loose water will leak up through the top around the ball and the faucet may not shut off properly. If the cam and packing adjustment is too tight the handle will be hard to move and the packing will have accelerated wear.